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Spatial coordinates *(X, Y, Z)* can be easily computed from
geographical ellipsoidal coordinates *(B, L, H)*, where *B*
is geographical latitude, *L* geographical longitude and *H*
is elliposidal height, as

X = (N + H) cos B cos L Y = (N + H) cos B sin L Z = (N(1-e^2) + H)sin B |

where
*N = a/sqrt(1 - e^2 sin^2 B)*
is the radius of curvature in the prime vertical, *e^2 = (a^2 -
b^2)/a^2* is the first eccentricity for the given rotational
ellipsoid (spheroid) with semi-major axis *a* and semi-minor
axis *b*.

In the case of coordiante transformation from *(X, Y, Z)*
to *(B, L, H)*, the longitude is given by the formula

tan L = Y / X. |

Now we can introduce

D = sqrt(X^2 + Y^2), |

so that the cartesian system become *(D, Z)*.
Coordinates *B* and *H* are then usually computed by
iteration with some starting value of *B_0*, for example

tan B_0 = Z/D/(1 - e^2), |

tan B = Z/D + N/(N+H) e^2 tan B, H = D / cos B = Z / sin B - N(1-e^2) |

B. R. Bowring described a closed formula(3) that is more effective and sufficiantly accurate and that is used in GNU Gama.

The centre of curvature *C* of the spheroid corresponding to
*P’* is the point

*(e^2 a cos^3 u, -e’^2 b sin^3 u))*,

where *e’^2 = (a^2 - b^2)/b^2* is second eccentricity and
*u* is the parametric latitude of the point
*P’, (1-e^2)N sin B = b sin u.*
Therefore

*tan B = (Z + e’^2 b sin^3 u) / (D - e^2 a cos^3 u)*.

This is clearly an iterative solution; but it has been found that this
formula is extremely accurate using the single first approximation for
*u* for the
*tan u = (Z/D)(a/b)*.
Maximum error in earth bound region is 3e-8 of
sexagesimal arc seconds (5e-7 millimetres); maximum is 0.0018” (0.1
millimetres) at height H = 2a.

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